A digital-data transmission system transmits digital data over a transmission media where recovery of the digital data is accomplished. During transmission, the digital data is conventionally converted to discrete-time constellation signals that are selected from a finite M-ary constellation alphabet. These constellation signals are subsequently converted to continuous-time transmission signals. Prior to, or during, transmission these signals may be distorted by nonlinear elements, for example a nonlinear power amplifier in the transmitter of a radio system. At a receiver, continuous-time received signals are converted to discrete-time receiver signals by frequency conversion from transmission frequencies to baseband, filtering, and periodic signal sampling. The receiver signals contain distorted-constellation signals and noise signals. Typically, the nonlinear element has a zero-memory nonlinearity but linear filtering prior to and after the nonlinear element produces dispersive nonlinear distortions. Consequently, a distorted-constellation signal depends in a nonlinear functional relationship on multiple successive constellation signals. Accordingly, the channel producing the receiver signals from the constellation signals can he represented as a discrete-time dispersive nonlinear channel.
In cancellation systems, such as successive-interference cancellation (SIC), the noise signal contains a desired signal. For example, in multiuser applications a SIC receiver can be used to cancel nonlinear-distorted interference, that is associated with a previously-demodulated stronger user, in order to demodulate the next weaker user. Accordingly, in cancellation systems, it is desirable to estimate the distorted-constellation signal and subtract the estimate from the receiver signal to obtain an estimate of the desired signal. These cancellation systems conventionally include a copy of the constellation signal sequence that is either reproduced from a known source or estimated in an earlier demodulation operation. The distorted-constellation signal estimate is found by characterizing the discrete-time dispersive nonlinear channel using the constellation signal copies as channel inputs. A best characterization minimizes the mean square difference between the receiver signals and the channel outputs. For this best characterization, the estimates of the distorted-constellation signals are the channel outputs.
Demodulation systems obtain the digital data from the receiver signals and the demodulation can be improved with estimates of the distorted-constellation signals. To obtain these estimates, one can use past binary-data decisions and hypotheses to generate source signals that are associated with the unknown constellation signals. The estimates are then found as described above in the cancellation system using the source signals as channel inputs,
Additionally there are channel identification applications where it is desirable to characterize the discrete-time dispersive nonlinear channel. Such a characterization can be accomplished from an estimation of the distorted-constellation signals given the constellation signals, as described above.
Distortions produced by a signal that traverses a nonlinear channel are often characterized by a Volterra series expansion. The Volterra series is a generalization of the classical Taylor series. See “Nonlinear System Modeling Based on the Wiener Theory”, Proceeding of the IEEE, vol. 69, no. 12, pp. 1557-1573, December 1981. U.S. Pat. No. 3,600,681 discloses a nonlinear equalizer based on a Volterra series expansion of nonlinear intersignal interference (NISI) in a data communication system. In “Adaptive Equalization of Channel Nonlinearities in QAM Data Transmission Systems”, D. D. Falconer, Bell System Technical Journal, vol. 57, No. 7, September 1978, [Falconer], the Volterra series for NISI is used in a passband decision feedback equalizer. This equalizer is adapted by adjusting the coefficients of the Volterra series expansion by a gradient algorithm. In Falconer, it was concluded that “the number of nonlinear terms . . . is potentially enormous” and that “the simulation results indicated that inclusion of a large number of nonlinear terms, . . . may be necessary.” The complexity of the Volterra series expansion for either voiceband telephone channels or satellite channels with nonlinear power amplifiers has been recognized in “Efficient Equalization of Nonlinear Communication Channels, W. Frank and U. Appel. 1997 IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. III, Apr. 21-24, 1997. [Frank]. In Frank, it is described that a decision feedback equalizer (DFE) uses a nonlinear structure that is a good approximation to the general Volterra filter but with reduced complexity. The nonlinear structure is based on an equivalent lowpass model of a 3rd order bandpass nonlinearity. Because this Volterra series approximation provided better improvements at higher signal-to-noise ratio, it is concluded in Frank that the Volterra approximation DFE is better suited to the voiceband telephone channel than radio communications.
Rather than provide compensation for nonlinear distortions at the receiver by using nonlinear equalizers, there are predistortion techniques that can be applied in the transmitter before the nonlinear channel. In “A Data Predistortion Technique with Memory for QAM Radio Systems”, IEEE Trans. Communications, Vol. 39, No. 2, February 1991, G. Karam and H. Sari, [Karam], explicit expressions are derived for the 3rd and 5th order inverse Volterra kernels. Karam also notes that the finite-order inverses grow “very rapidly” with the Volterra order p and the discrete-time signal memory span K. These small order/memory span Volterra inverses are compared in Karam with a lookup memory encoder (referred to as “global compensation” in Karam) that predistorts each possible discrete-time signal data value such that at the discrete-time channel output the center of gravity of the received points is in the correct position in the discrete-time signal constellation. The RAM implementation of the lookup memory encoder requires K log2 M address bits where M is the modulation alphabet size. By using a rotation technique based on the center discrete-time signal in the memory span, the number of address bits can be reduced in M-ary QAM by two because of quadrature-phase symmetry. For a given memory span and a practical number of address bits, it is described in Karam that the lookup memory encoder outperforms the Volterra inverse predistortion. However, Karam does not describe a technique for initializing and adapting the lookup memory encoder in the presence of additive noise. Unfortunately, the preamble length for initialization of a predistortion lookup memory encoder can be excessively large. The preamble length is on the order of AMK−1 discrete-time signals where A is the averaging time to make the additive noise small compared to an acceptable level of residual distortion. A typical averaging time of 100 discrete-time signals for 8 PSK with K=5 would require a preamble of over 400,000 discrete-time signals. This difficulty with initialization and adaptation of distortion compensation systems using lookup table techniques is also noted in “modeling and Identification of a Nonlinear Power-Amplifier with Memory for Nonlinear Digital Adaptive Pre-Distortion”, Proceedings of the SPAWC Workshop, 15-18.6.2003, Rome Italy, by Aschbacher et al, [Aschbacher]. Also recognizing the slow convergence and large number of coefficients in the Volterra series expansion, it is suggested in Aschbacher to identify a nonlinear power amplifier by a simplified Wiener-model consisting of a linear filter followed by a zero-memory nonlinearity. An adaptive Least Means Squares algorithm is used to adapt and track parameters in the linear filter and the zero-memory nonlinearily to minimize the mean square error between the sampled data output of the nonlinear power amplifier and the simplified Wiener-model. This minimization is over the signal bandwidth rather than the smaller discrete-signal bandwidth and the minimization does not include receiver filtering contributions to the nonlinear intersignal interference. As a result interference cancellation with the Aschbacher identification model would not be as effective as a technique that is receiver based and minimizes a mean square error die received discrete-time signal values.
Accordingly, there is a need at a receiver terminal in certain digital-data communication systems with discrete-time signals that traverse a nonlinear-dispersive channel for estimation of the received distorted signal. It would be desirable to utilize nonlinear techniques that provide faster convergence of the nonlinear series expansion and better performance than prior art systems based on conventional Volterra series expansion techniques. Additionally, it would be desirable that these nonlinear techniques can be initialized and adapted to changing conditions more effectively than prior art lookup memory techniques.